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Mapping without the sun
Zhang, Jixian

outliers O k . We use a cyclic coordinate optimization procedure
solve all the unknowns. The HR image, registration parameters
and outliers are found in an alternate manner given the two
others, respectively. A block diagram of the whole optimization
procedure is shown in Figure 1.
4.1 Registration
To solve the registration parameters m k , we first expand the
cost function (16) by Taylor formula ( For expression
convenience, we neglect the subscript k) :
£(m)*£(m”)-K^^) r (Am) + -(Am) r /7"(Am) (17)
dm" 2
dE(m)_ an( j Yjn ¿ enote t h e
where Yl is the iteration number,
dm n
gradient matrix and Hessian matrix of E(m) atm” respectively:
®W=2(/-)'Oy 08)
dm" *
H" = 2(7” ) r O n k J n + 2£ a;"0;( 1 9)
In (18) and (19), r” is the residual vector that is equal
n dr n
t0 Sk ~Kk DB k M kZ-h 0 J ,J = denotes the gradient
matrix of r" , and H" is the Hessian matrix of r". For small
r" , we have the following approximations:
H" ~ 2(J") T O k J n . Thus, differentiating (6) with respect to
Am and setting the result equal to zero, after some
manipulation, we have
(,J") r O n k J n (Am) = -(J n ) T O k r" (20)
Am = [(J n fOlJ" r 11 H J" f O n k r n ] (21)
Then, the parameter vector can be updated by
m'" ! = m n + Am (22)
r = -\ k MlB T k D T 0 T k {g k -\ k DB k M k z-\ k I) + Xr (26)
where /* is the derivative of the regularization term that can be
solved on a pixel-by-pixel basis. Thus, the HR image is solved
by employing the successive approximations iteration
2" +1 =z"-fir" (27)
where Yl is the iteration number, and J3 is the step size which
can be solved by
/? =
(■r n fr"
(r") T (V 2 E)r"
where V 2 E is the Hessian matrix of the objective
function E(z, m, O).
(a) (b)
(c) (d)
4.2 Outlier Detection
To detect the outliers, we employ two criteria. The first is a
geometrical criterion that requires the predicted location of a
pixel using the current motion parameters is still in the image
field. The horizontal and vertical predicted locations are judged
by equation (23) and (24) respectively.
0 < a 0 + a x x x + a 2 y { < (23)
0 < b 0 + b x x x + b 2 y x < N 2 (24)
Here, N x and N 2 are respectively the horizontal and vertical
size of the observed image. The second criterion is a
photometrical one. Define f k = h x k DB k M k z~ h 0 k I , the
photometrical criterion is
I gk ( x > y) ~ fk (*> y)\ where cr k (x, y) is the standard deviation at site (x, y) of the
kth image g k , and d in the right hand side acts as a scale
4.3 SR Reconstruction
The steepest descent optimization is used to solve the HR
image z ■ Differentiating (16) with respect to z , we have
(g) (h)
Figure, (a)-(f) Images respectively captured on 28/12/2003.
30/12/2003, 01/01/2004, 04/01/2004, 06/01/2004 and
08/01/2004, (g)
reconstmcted ima
The proposed alg
images which wer
6, 8 January 2004
the Satellite Ren
University. We cl
algorithm indeper
the two regions a:
Figures 3(a)-(f)-
Figure 3. (a)-(f) Irr
30/12/2003, 01/(
08/01/2004, (g)
reconstructed imag
In the experiment
as: ^ = 1 , X = 0.1
kernel with unit \
employed. We assi